Stochastic multi-scale modeling and simulation for nonlinear thermo-mechanical problems of composite materials with complicated random microstructures remains a challenging issue. In this paper, we develop a novel statistical higher-order multi-scale (SHOMS) method for nonlinear thermo-mechanical simulation of random composite materials, which is designed to overcome limitations of prohibitive computation involving the macro-scale and micro-scale. By virtue of statistical multi-scale asymptotic analysis and Taylor series method, the SHOMS computational model is rigorously derived for accurately analyzing nonlinear thermo-mechanical responses of random composite materials both in the macro-scale and micro-scale. Moreover, the local error analysis of SHOMS solutions in the point-wise sense clearly illustrates the crucial indispensability of establishing the higher-order asymptotic corrected terms in SHOMS computational model for keeping the conservation of local energy and momentum. Then, the corresponding space-time multi-scale numerical algorithm with off-line and on-line stages is designed to efficiently simulate nonlinear thermo-mechanical behaviors of random composite materials. Finally, extensive numerical experiments are presented to gauge the efficiency and accuracy of the proposed SHOMS approach.

翻译：随机多尺度建模与模拟在具有复杂随机微结构的复合材料非线性热-力学问题中仍是一个具有挑战性的课题。本文发展了一种新颖的统计高阶多尺度方法（SHOMS），用于随机复合材料的非线性热-力学模拟，旨在克服涉及宏观尺度和微观尺度的计算量过大的局限性。通过统计多尺度渐近分析和泰勒级数方法，严格推导了SHOMS计算模型，可精确分析随机复合材料在宏观和微观尺度上的非线性热-力学响应。此外，基于点态意义的SHOMS解局部误差分析，清晰阐明了在SHOMS计算模型中建立高阶渐近修正项对于保持局部能量与动量守恒的关键必要性。随后，设计了离线和在线阶段的时空多尺度数值算法，以高效模拟随机复合材料的非线性热-力学行为。最后，通过大量数值实验评估了所提出的SHOMS方法的效率和精度。